the lognormal distribution

Some quantities cannot go negative and span several orders of magnitude — incomes, stock prices, city populations, the size of biological cells. Their histograms are not symmetric bells but lopsided humps with a long right tail. Often the right model is the lognormal distribution: a variable whose LOGARITHM, not the value itself, is normally distributed.

Say X is lognormal if log(X) ~ Normal(mu, sigma^2). Equivalently, X = e^Y where Y is normal. The trick to working with it is exactly that: take logs, and the problem becomes an ordinary normal problem. Whereas a normal arises from adding many small independent effects, a lognormal arises from MULTIPLYING them — and multiplying becomes adding once you take logs (log of a product is the sum of logs), so the central limit theorem acting on the logs delivers a normal there, hence a lognormal for X. Note its mean is NOT e^mu; the right-skew pushes the mean above the median, with E[X] = e^(mu + sigma^2/2).

Lognormals are everywhere growth is proportional — money compounding by a random percentage each year, particle sizes after repeated random splits, file sizes on a disk. The key honest points: only positive values are possible, the distribution is right-skewed so mean, median, and mode differ, and you must be careful never to confuse the parameters mu and sigma (which describe log(X)) with the actual mean and standard deviation of X itself.